TeXipedia

exists

Denotes logical existence, indicating that there is at least one element satisfying a given condition.

Overview

Essential in mathematical logic, set theory, and formal proofs where existence claims need to be expressed precisely.

  • Commonly used in theorem statements and mathematical definitions
  • Frequently paired with quantifiers and predicates in logical expressions
  • Appears extensively in abstract algebra and analysis when describing properties of sets
  • Often combined with other logical symbols to construct complex mathematical statements
  • Particularly important in existence proofs and constructive mathematics

Examples

Expressing the existence of a solution to an equation

∃x∈R:x2+2x+1=0\exists x \in \mathbb{R} : x^2 + 2x + 1 = 0 
\exists x \in \mathbb{R} : x^2 + 2x + 1 = 0

Stating the existence of a unique element in set theory

∃!x∈A:f(x)=0\exists! x \in A : f(x) = 0 
\exists! x \in A : f(x) = 0

Expressing existence in logical notation

∃x∀y(P(x)  ⟹  Q(y))\exists x \forall y (P(x) \implies Q(y)) 
\exists x \forall y (P(x) \implies Q(y))